Abstract: The Stephenson Ⅲ six-bar linkage consists of the planar four-bar linkage and the external dyad.The branches are the domains on the coupler curves separated by two limit points or limit and turning points.The circuits are the coupler curve or domains separated by two turning points.In this paper,it is shown that these branches and circuits can be mapped on four number lines discriminated by the sign of the determinant of the Jacobian matrix and the sign of the cosine of the relative angular displacement of the connecting link.